Core-Nilpotent decomposition and new generalized inverses of finite potent endomorphisms

ABSTRACT

The aim of this work is to prove the existence and uniqueness of a core-nilpotent decomposition of finite potent endomorphisms on arbitrary vector spaces. This decomposition generalized the well-known core-nilpotent decomposition of complex $(n\times n)$-matrices. As an application we offer the definition and properties of the CMP inverse and of a new CR inverse for these linear maps. In particular, from the results of this work it is possible to compute the core-nilpotent decomposition, the CMP and the CR inverses of some infinite matrices.

COMMUNICATED BY:

• R. B. Bapat

Keywords:

• Finite potent endomorphism
• Core-Nilpotent decomposition
• CMP Inverse
• CR inverse

2010 Mathematics Subject Classifications:

• 15A09
• 15A03
• 15A04

Acknowledgements

The author would like to thank the anonymous reviewer for his/her valuable comments to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work is partially supported by the Spanish Government research (Ministerio de Economía y Competitividad) project no. MTM2015-66760-P and the Regional Government of Castile and Leon research (Consejería de Educación, Junta de Castilla y León) project no. SA030G18.